Free examples and use-cases:   rpact vignettes
rpact: Confirmatory Adaptive Clinical Trial Design and Analysis

# Summary

This R Markdown document shows how to analyse and interpret multi-arm designs for testing proportions with rpact.

# 1 Introduction

This vignette provides examples of how to analyse a trial with multiple arms and a binary endpoint. It shows how to calculate the conditional power at a given stage and to select/deselect treatment arms. For designs with multiple arms, rpact enables the analysis using the closed combination testing principle. For a description of the methodology please refer to Part III of the book “Group Sequential and Confirmatory Adaptive Designs in Clinical Trials” (Wassmer and Brannath, 2016).

Suppose the trial was conducted as a multi-arm multi-stage trial with three active treatments arms and a control arm when the trial started. In the interim stages, it should be possible to de-select treatment arms if the treatment effect is too small to show significance - assuming reasonable sample size - at the end of the trial. This should hold true even if a certain sample size increase was taken into account. The endpoint is a failure and it is intended to test each active arm against control. This is to test the hypotheses $H_{0i}:\pi_{\text{arm}i} = \pi_\text{control} \qquad\text{against} \qquad H_{1i}:\pi_{\text{arm}i} < \pi_\text{control}\;, \;i = 1,2,3\,,$ in the many-to-one comparisons setting. That is, it is intended to show that the failure rate is smaller in active arms as compared to control and so the power is directed towards negative values of $$\pi_{\text{arm}i} - \pi_\text{control}$$.

# 2 Create the design

library(rpact)
packageVersion("rpact") # version should be version 3.0 or later
## [1] '3.3.2'

In rpact, we first have to select the combination test with the corresponding stopping boundaries to be used in the closed testing procedure. We choose a design with critical values within the Wang & Tsiatis $$\Delta$$-class of boundaries with $$\Delta = 0.25$$. Planning two interim stages and a final stage, assuming equally sized stages, the design is defined through

designIN <- getDesignInverseNormal(
kMax = 3, alpha = 0.025,
typeOfDesign = "WT", deltaWT = 0.25
)
kable(summary(designIN))

Sequential analysis with a maximum of 3 looks (inverse normal combination test design)

Wang & Tsiatis Delta class design (deltaWT = 0.25), one-sided overall significance level 2.5%, power 80%, undefined endpoint, inflation factor 1.0544, ASN H1 0.8202, ASN H01 0.9966, ASN H0 1.0489.

Stage 1 2 3
Information rate 33.3% 66.7% 100%
Efficacy boundary (z-value scale) 2.741 2.305 2.083
Stage Levels 0.0031 0.0106 0.0186
Cumulative alpha spent 0.0031 0.0124 0.0250
Overall power 0.1400 0.5262 0.8000

This definition fixes the weights in the combination test which are the same over the three stages. This is a reasonable choice although the amount of information seems to be not the same over the stages (see Wassmer, 2010).

# 3 Analysis

## 3.1 First stage

In each treatment and the control arm, subjects were randomized such that around 40 subjects per arm will be observed. Assume that the following actual sample sizes and failures in the control and the three experimental treatment arms were obtained for the first stage of the trial:

Arm n Failures
Active 1 42 7
Active 2 39 8
Active 3 38 14
Control 41 18

These data are defined as an rpact dataset with the function getDataset() for the later use in getAnalysisResults() through

dataRates <- getDataset(
events1      =  7,
events2      =  8,
events3      = 14,
events4      = 18,
sampleSizes1 = 42,
sampleSizes2 = 39,
sampleSizes3 = 38,
sampleSizes4 = 41
)

That is, you can use the getDataset() function in the usual way and simply extend it to the multiple treatment arms situation. Note that the arm with the highest index always refers to the control group. For the control group, specifically, it is mandatory to enter values over all stages. As we will see below, it is possible to omit information of de-selected active arms.

Using

results <- getAnalysisResults(
design = designIN, dataInput = dataRates,
directionUpper = FALSE
)
kable(summary(results))

one obtains the test results for the first stage of this trial (note the directionUpper = FALSE specification that yields small $$p$$-values for negative test statistics):

Multi-arm analysis results for a binary endpoint (3 active arms vs. control)

Sequential analysis with 3 looks (inverse normal combination test design). The results were calculated using a multi-arm test for rates (one-sided), Dunnett intersection test, normal approximation test. H0: pi(i) - pi(control) = 0 against H1: pi(i) - pi(control) < 0.

Stage 1 2 3
Fixed weight 0.577 0.577 0.577
Efficacy boundary (z-value scale) 2.741 2.305 2.083
Cumulative alpha spent 0.0031 0.0124 0.0250
Stage level 0.0031 0.0106 0.0186
Cumulative effect size (1) -0.272
Cumulative effect size (2) -0.234
Cumulative effect size (3) -0.071
Cumulative treatment rate (1) 0.167
Cumulative treatment rate (2) 0.205
Cumulative treatment rate (3) 0.368
Cumulative control rate 0.439
Stage-wise test statistic (1) -2.704
Stage-wise test statistic (2) -2.233
Stage-wise test statistic (3) -0.639
Stage-wise p-value (1) 0.0034
Stage-wise p-value (2) 0.0128
Stage-wise p-value (3) 0.2615
Adjusted stage-wise p-value (1, 2, 3) 0.0095
Adjusted stage-wise p-value (1, 2) 0.0066
Adjusted stage-wise p-value (1, 3) 0.0066
Adjusted stage-wise p-value (2, 3) 0.0239
Overall adjusted test statistic (1, 2, 3) 2.346
Overall adjusted test statistic (1, 2) 2.480
Overall adjusted test statistic (1, 3) 2.480
Overall adjusted test statistic (2, 3) 1.980
Overall adjusted test statistic (1) 2.704
Overall adjusted test statistic (2) 2.233
Overall adjusted test statistic (3) 0.639
Test action: reject (1) FALSE
Test action: reject (2) FALSE
Test action: reject (3) FALSE
Conditional rejection probability (1) 0.2647
Conditional rejection probability (2) 0.1708
Conditional rejection probability (3) 0.0202
95% repeated confidence interval (1) [-0.541; 0.038]
95% repeated confidence interval (2) [-0.514; 0.089]
95% repeated confidence interval (3) [-0.384; 0.259]
Repeated p-value (1) 0.0519
Repeated p-value (2) 0.0948
Repeated p-value (3) 0.4568

Legend:

• (i): results of treatment arm i vs. control arm
• (i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

First of all, at the first interim no hypothesis can be rejected with the closed combination test. This is seen from the test action: reject (i) variable. It is remarkable, however, that the $$p$$-value for the comparison of treatment arm 1 against control (p = 0.0034) is quite small and even the $$p$$-value for the global intersection is (p(1, 2, 3) = 0.0095) is not too far from showing significance. It is important to know that, by default, the Dunnett many-to-one comparison test for binary data is used as the test for the intersection hypotheses, and the approximate pairwise score test (which is the signed square root of the $$\chi^2$$ test) is used for the calculation of the separate $$p$$-values. Note that in this presentation the intersection tests for the whole closed system of hypotheses is provided such that the closed test can completely be reproduced.

The repeated $$p$$-values (0.0519, 0.0948, and 0.4568, respectively) precisely correspond with the test decision meaning that a repeated $$p$$-value is smaller or equal to the overall significance level (0.025) if and only if the corresponding hypothesis can be rejected at the considered stage. This direct correspondence is not generally true for the repeated confidence intervals (i.e., they can contain the value zero although the null hypothesis can be rejected), but it is true for the situation at hand. The repeated confidence intervals can be displayed with the plot(, type = 2) command by

plot(results, type = 2)